Can every one variable equation be solved without graphing?

Question

Can every one variable equation be solved without graphing?

How would you solve the following without graphing:

$$3y + 4\sqrt{1-y^2} = 2$$

Answer 1

We see $$3y-2=-4\sqrt{1-y^2}$$ so $$(3y-2)^2=16-16y^2$$ therefore $$9y^2-12y+4=16-16y^2$$ rearranging gives $$25y^2-12y-12=0$$ And so the solutions are $$y_0,y_1=\frac{12}{50}\pm\frac{1}{50}\sqrt{144+1200}$$ and simplifying yields $$y_0,y_1=\frac{6\pm4\sqrt{21}}{25}$$ Checking these solutions will give us the unique solution $$y_0=\frac{6-4\sqrt{21}}{25}$$

Answer 2

One canonical example of an equation of one variable that can't be solved without graphing is $$ x^2=e^x.$$ Try it for yourself if you'd like.